The Relations between Residuated Frames and Residuated Connections

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind&#8722;MacNeille completion, we introduce <i>R</i>-<i>R</i> (resp. <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mi>R</mi> </mrow> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mi>R</mi> </mrow> </semantics> </math> </inline-formula>) embedding maps and <i>R</i>-<i>R</i> (resp. <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mi>R</mi> </mrow> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mi>R</mi> </mrow> </semantics> </math> </inline-formula>) frame embedding maps.

Abstract licence: CC BY

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